Mathematics for the interested outsider

2008 Abel Prize

As Chris Hillman just pointed out in a comment, the 2008 Abel Prize went to John Griggs Thompson and Jacques Tits “for their profound achievements in algebra and in particular for shaping modern group theory”. The comment went on my recent throwaway post about the 7x7x7 Rubik’s Cube, but a more appropriate one might have been this one from over a year ago, in which I discuss the Feit-Thompson theorem in passing.

Incidentally, I think I’ve met both of the winners. Tits I’m sure of, since I tried and failed horribly to take a short course he gave at Yale on “buildings”. Thompson I believe showed up for Walter Feit’s memorial, but I could be wrong about that. I wish I could say I was particularly close to one or the other, but I suppose that will have to wait until Adams and Vogan win the prize.

Advertisements

Share this:

Like this:

Related

If you take a stab at trying to write a few paragraphs of background for the award to Thompson, I might take a stab at writing a few paragraphs of background for the award to Tits. Or, since John Baez has written about the work of both Thompson and Tits in past Weeks, we can both wait for the next installment of TWF.

At some point I plan to post something on the n-Category Café on an approach to buildings via enriched category theory. There is a slogan to the effect that buildings are certain kinds of “metric spaces” where the distance function is valued in a Coxeter group; the approach is to take that seriously, in the context of Lawvere’s approach to metric spaces via enriched category theory.

Todd,
a considerable portion of the theory of building has been built using the slogan you mention, so you don’t really need to advocate taking it seriously…
The axiomatics there is however a bit intimidating, so perhaps indeed the category theory might clarify something.

The axiomatics there is however a bit intimidating, so perhaps indeed the category theory might clarify something.

I want to hammer on this a bit, actually. People have gone back and forth over what the hell category theory is good for. Here it is: it can provide an organizing viewpoint, clearing up what is actual data and what is bookkeeping. The categorical viewpoint sweeps these ornate axiomatics into tidy little piles that we can actually wrap our heads around.

Dima, what I meant was that the notion of building, involving the yoga of reduced words in Coxeter groups, doesn’t seem at first glance to be based on concepts which have recognizable homologues for ordinary metric spaces. In fact, in some treatments, e.g., in Ronan’s book The Structure of Spherical Buildings, the meaning of something so basic as “triangle inequality” may not be immediately clear and precise. (Yes, there is of course the Bruhat order, but it is not preserved by translation in the Coxeter group in the same way that the order of the reals is preserved by additive translation, so for the non-experts the analogy may need a little work before it becomes fully satisfying).

Anyway, a categorically minded person might like to find the right general notion which in one specialization yields precisely Tits buildings, and in another yields metric spaces satisfying certain geodesic properties… the context I have in mind (which Jim Dolan and I worked out together) involves some nice enriched category theory and categorical logic.

I was planning to write about more fundamental background, trying to explain the basic ideas of Klein’s vision, explaining how one deduce from the stabilizer lattice information about how “geometrical elements” combine, for example in finite projective planes.

There is another, less well-known, theory of diagram geometries, modeled after Tits’ “Local approach to buildings”. It includes buildings as a particular case.

Instead of requiring that the “rank 2 residues” (that is, what do you get if you take the subsets of maximal flags that conicide everywhere, except two types) are generalised k-gons, one allows a wider class of objects (or one can, say, require that this class has particular things there, apart from the generalised k-gons, e.g., the incidence system of vertices and edges of the Petersen graph, or, e.g., the points and the lines of an affine plane)

Certainly, absence of apartments in such more general things make classification much more difficult than in the building case. On the other hand one can e.g. talk about “geometric hyperplanes” of buildings, or have a theory characterising some sporadic finite simple groups.

Chris, where were you planning to write about this? Would you consider writing a guest post for the n-Category Café?

I think John Baez wanted me to hold off on writing a guest post about buildings (as part of the Geometric Representation Theory Seminar he and Jim Dolan are running), but since there is some interest in it now, maybe I’ll ask him again. Otherwise I might clean up and expand on some notes which have meanwhile appeared under my name on his website. (I’d have to look to remind myself exactly what’s in them now!) But perhaps it might make sense for you and me (Chris) to put our heads together — what you’re planning sounds like it would fit in well with what I’d say.

Since I already have an account at Ars Mathematica, I am drafting a post on Kleinian geometry (as motivating background for the theory of buildings), but I can already see that it may not be possible under WordPress to upload the figures I planned showing stabilizer-fixset lattices and simple geometric sketches (for some examples I planned to discuss). This would place strong constraints on the kind of post I could attempt to write.

Jacques’s software supports math formatting has some other advantages, but unfortunately MathML is incompatible with my own system (wrong libraries, not just missing fonts), so unfortunately I couldn’t take advantage of the nicer features at the Cafe; as at Ars Mathematica I would be limited to figureless equationless text. (If I am wrong in thinking that WordPress doesn’t support even small xfig figures uploaded as jpegs or whatever, please tell me!) Better than either WordPress or Jacques software for my purposes would be VBulletin (that’s what they use at Physics Forums and many similar bulletin boards), which supports reasonably nice if rudimentary mathematical formatting and also allows for small figures. Better by far would be WikiMedia with permissions set for use as a website authorship tool rather than a wiki, since this allows much greater flexibility in using figures, tables, and so on.

Despite these limitations I plan to just forge ahead with a draft and see how it comes out. I hope you, Todd, will go ahead with your post on buildings ASAP! FYI, I am planning to try to get across the most important elements of Klein’s vision, especially Galois duality between fixsets and (pointwise) stabilizers, and to discuss some concrete examples to show how geometrical information can be inferred from the stabilizer lattice. The examples I have in mind are finite projective spaces under PGL(d+1,F) and actions by the symmetry group of simplices and d-cubes. My idea is that this could segue into a discussion of finite analogues of more “rigid” geometries like affine, sympletic, euclidean, which is getting close to motivation for buildings, but to talk about that I’d probably need to first talk about homogeneous spaces (as in Lie theory). The theme I have in mind is that Klein proposed that studying the symmetry group of a geometry gives much insight into the geometry, and conversely, constructing the “geometry” corresponding to a group (rather, a group action) gives insight into the nature of the group. He said quite a bit about the first half but less about the second. Buildings can be viewed as partially realizing the second half of Klein’s program.

And John, since you’ve written about Rubik’s Cube, which I have used in this context as an example of an “unscrambling problem” which can be formulated in “information theoretic” terms, perhaps I can continue that thread here.

Dima, I’m wondering if the reason that Ars Mathematica doesn’t support LaTeX is that it’s using the WordPress software, but not the hosting. Thus the interface might be different and it may be a lot more awkward to upload pictures to the host for him.

John, I stand corrected. I haven’t understood the issue.
On the other hand, img html tag is quite easy to learn to use (imho, it’s easier than latex’s picture environments), while images can be placed most anyplace on the web after all…

John, off-topic, I wonder how the URL I gave to sign my posts seem to follow me here. Now I removed it from the places under my control; I wonder whether it’s hashed in some way in the guts of your blog…

OK, now it seems to have changed to what I intended. So it’s by default not the name of my wordpress blog, but the URL specified in the contact info of the wordpress account…
Once again, apologies for offtopic here.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.